Logic Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! ===Extended=== Extended logics are logical systems that accept the basic principles of classical logic. They introduce additional symbols and principles to apply it to fields like [[metaphysics]], [[ethics]], and [[epistemology]].{{sfnm |1a1=Bunnin |1a2=Yu |1y=2009|1p=[https://books.google.com/books?id=M7ZFEAAAQBAJ&pg=PA179 179] |2a1=Garson |2y=2023 |2loc=[https://plato.stanford.edu/entries/logic-modal/ Introduction]}} ====Modal logic==== {{Main|Modal logic}} [[Modal logic]] is an extension of classical logic. In its original form, sometimes called "alethic modal logic", it introduces two new symbols: <math>\Diamond</math> expresses that something is possible while <math>\Box</math> expresses that something is necessary.{{sfnm|1a1=Garson|1y=2023|2a1=Sadegh-Zadeh|2y=2015|2p=983}} For example, if the formula <math>B(s)</math> stands for the sentence "Socrates is a banker" then the formula <math>\Diamond B(s)</math> articulates the sentence "It is possible that Socrates is a banker".{{sfn |Fitch |2014 |p=17}} To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences. One rule of inference states that, if something is necessary, then it is also possible. This means that <math>\Diamond A</math> follows from <math>\Box A</math>. Another principle states that if a proposition is necessary then its negation is impossible and vice versa. This means that <math>\Box A</math> is equivalent to <math>\lnot \Diamond \lnot A</math>.{{sfnm|1a1=Garson|1y=2023|2a1=Carnielli|2a2=Pizzi|2y=2008|2p=3|3a1=Benthem}} Other forms of modal logic introduce similar symbols but associate different meanings with them to apply modal logic to other fields. For example, [[deontic logic]] concerns the field of ethics and introduces symbols to express the ideas of [[obligation]] and [[Permission (philosophy)|permission]], i.e. to describe whether an agent has to perform a certain action or is allowed to perform it.{{sfn |Garson |2023}} The modal operators in [[Temporal logic|temporal modal logic]] articulate temporal relations. They can be used to express, for example, that something happened at one time or that something is happening all the time.{{sfn |Garson |2023}} In epistemology, [[epistemic modal logic]] is used to represent the ideas of [[Knowledge|knowing]] something in contrast to merely [[Belief|believing]] it to be the case.{{sfn |Rendsvig |Symons |2021}} ====Higher order logic==== {{Main|Higher-order logic}} [[Higher-order logic|Higher-order logics]] extend classical logic not by using modal operators but by introducing new forms of quantification.{{sfnm|1a1=Audi|1loc=Philosophy of logic|1y=1999b|2a1=Väänänen|2y=2021|3a1=Ketland|3y=2005|3loc=Second Order Logic}} Quantifiers correspond to terms like "all" or "some". In classical first-order logic, quantifiers are only applied to individuals. The formula {{nowrap|"<math>\exists x (Apple(x) \land Sweet(x))</math>"}} (''some'' apples are sweet) is an example of the [[Existential quantification|existential quantifier]] {{nowrap|"<math>\exists</math>"}} applied to the individual variable {{nowrap|"<math>x</math>"}}. In higher-order logics, quantification is also allowed over predicates. This increases its expressive power. For example, to express the idea that Mary and John share some qualities, one could use the formula {{nowrap|"<math>\exists Q (Q(Mary) \land Q(John))</math>"}}. In this case, the existential quantifier is applied to the predicate variable {{nowrap|"<math>Q</math>"}}.{{sfnm|1a1=Audi|1loc=Philosophy of logic|1y=1999b|2a1=Väänänen|2y=2021|3a1=Daintith|3a2=Wright|3y=2008|3loc=[https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/predicate-calculus Predicate calculus]}} The added expressive power is especially useful for mathematics since it allows for more succinct formulations of mathematical theories.{{sfn |Audi |loc=Philosophy of logic |1999b}} But it has drawbacks in regard to its meta-logical properties and ontological implications, which is why first-order logic is still more commonly used.{{sfnm|1a1=Audi|1loc=Philosophy of logic|1y=1999b|2a1=Ketland|2y=2005|2loc=Second Order Logic}} Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page